International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 27-29

Section 1.1.4.10.6. Independent components of the matrix associated with a fourth-rank tensor according to the following point groups

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.10.6. Independent components of the matrix associated with a fourth-rank tensor according to the following point groups

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1.1.4.10.6.1. Triclinic system, groups [{\bar 1}], [1]

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[Scheme scheme67]

1.1.4.10.6.2. Monoclinic system

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Groups [2/m], 2, m, twofold axis parallel to [Ox_{2}]: [Scheme scheme68]

1.1.4.10.6.3. Orthorhombic system

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Groups [mmm], [2mm], 222: [Scheme scheme69]

1.1.4.10.6.4. Trigonal system

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  • (i) Groups 3, [{\bar 3}]: [Scheme scheme70a] [Scheme scheme71] where ⊖ is a component numerically equal but opposite in sign to the heavy dot component to which it is linked; ⊕ is a component equal to twice the heavy dot component to which it is linked; ⊙ is a component equal to minus twice the heavy dot component to which it is linked; ⊗ is equal to [1/2(p_{11} - p_{12})], [(\pi_{11} - \pi_{12})], [2(Q_{11} - Q_{12})], [1/2(c_{11} - c_{12})] and [2(s_{11} - s_{12})], respectively.

  • (ii) Groups 32, [3m], [{\bar 3}m]: [Scheme scheme72] [Scheme scheme73] with the same conventions.

1.1.4.10.6.5. Tetragonal system

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  • (i) Groups 4, [{\bar 4}] and [4/m]: [Scheme scheme74]

  • (ii) Groups 422, [4mm], [{\bar 4}2m] and [4/mm]: [Scheme scheme75]

1.1.4.10.6.6. Hexagonal system

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  • (i) Groups 6, [{\bar 6}] and [6/m]: [Scheme scheme76] [Scheme scheme77]

  • (ii) Groups 622, [6mm], [{\bar 6}2m] and [6/mm]: [Scheme scheme78]

1.1.4.10.6.7. Cubic system

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  • (i) Groups 23 and [3m]: [Scheme scheme79]

  • (ii) Groups 432, [{\bar 4}3m] and [m{\bar 3}m]: [Scheme scheme80]

1.1.4.10.6.8. Spherical system

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For all tensors [Scheme scheme81]








































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